Incremental Gradient, Subgradient, and Proximal Methods for Convex Optimization A Survey

• Published in: Optimization for Machine Learning p. 85
• Main Author: Dimitri P. Bertsekas
• Format: Book Chapter
• Language: English
• Published: United States The MIT Press 30.09.2011; MIT Press
• Subjects: Applied mathematics; Applied sciences; Applied statistics; Artificial Intelligence; Calculus; Component functions; Computer Science; Cost functions; Data analysis; Differential calculus; Directional derivatives; Economic costs; Economic costs and benefits; Economics; Expected values; Gradient method; Inferential statistics; Information analysis; Information science; Least squares; Linear algebra; Linear regression; Machine Learning & Neural Networks; Mathematical differentiation; Mathematical expressions; Mathematical functions; Mathematical optimization; Mathematical procedures; Mathematical vectors; Mathematics; Measuring instruments; Microeconomics; Optimal solutions; Pure mathematics; Regression analysis; Scalars; Sensors; Statistics; Technology; Vector analysis; Vector operations; Vector valued functions
• ISBN: 026201646X; 9780262016469
• DOI: 10.7551/mitpress/8996.003.0006

We consider optimization problems with a cost function consisting of a large number of component functions, such as minimize$\sum\limits_{i = 1}^m {{f_i}(x)} $subject to$x \in X$, (4.1) where${f_i}:{R^n} \mapsto R$, i = 1 , . . . ,mare real-valued functions, andXis a closed convex set.¹ We focus on the case where the number of componentsmis very large, and there is an incentive to use incremental methods that operate on a single component${f_i}$at each iteration, rather than on the entire cost function. If each incremental iteration tends to make reasonable progress in some “average” sense, then, depending